Article
Mathematics
Martin Cech, Kaisa Matomaki
Summary: We study the non-vanishing property of Dirichlet L-functions at the central point, assuming the existence of an exceptional Dirichlet character. In particular, we prove that if ψ is a real primitive character modulo D, with L(1, ψ) << (log D)(-25-ε), then for any prime q in [D-300, D-O(1)], L(1/2, χ) is not equal to 0 for almost all Dirichlet characters χ (mod q).
MATHEMATISCHE ANNALEN
(2023)
Article
Mathematics
Jennifer Brooks, Megan Dixon, Michael Dorff, Alexander Lee, Rebekah Ottinger
Summary: In this paper, the authors investigated the variation of the number of zeros of a one-parameter family of harmonic trinomials with a real parameter. They also explored the number of zeros for convex combinations of members in these families. The harmonic analog of Rouche's theorem served as the main tool to prove these results.
Article
Mathematics, Applied
Louis Gass
Summary: We study the variance asymptotics of real zeros of trigonometric polynomials with random dependent Gaussian coefficients and find that, under mild conditions, they have the same asymptotic behavior as in the independent framework. Furthermore, our proof goes beyond this framework and explicitly determines the variance asymptotics of various random Gaussian process models. Our approach relies on the intrinsic properties of the Kac-Rice density, which allows for a concise and elegant proof of the variance asymptotics.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2023)
Article
Physics, Multidisciplinary
Boris Hanin, Steve Zelditch
Summary: We prove that smooth Wigner-Weyl spectral sums at an energy level E exhibit Airy scaling asymptotics across the classical energy surface sigma( E ). This extends the authors' previous proof for the isotropic harmonic oscillator to all quantum Hamiltonians of the form -PLANCK CONSTANT OVER TWO PI (2)Delta + V, where V is a confining potential with at most quadratic growth at infinity. The main tools used are the Herman-Kluk initial value parametrix for the propagator and the Chester-Friedman-Ursell normal form for complex phases with a one-dimensional cubic degeneracy. This provides a rigorous account of the Airy scaling asymptotics of spectral Wigner distributions observed by Berry, Ozorio de Almeida, and other physicists.
JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
(2022)
Article
Mathematics
Hung M. Bui, Kyle Pratt, Alexandru Zaharescu
Summary: Assuming the existence of exceptional characters, we have proven that at least fifty percent of the central values of the Dirichlet L-functions are nonzero, and for most cases, the function has at most a simple zero at s=1/2.
MATHEMATISCHE ANNALEN
(2021)
Article
Mathematics, Applied
Daniel Hu, Ikuya Kaneko, Spencer Martin, Carl Schildkraut
Summary: In this paper, we provide a new unconditional proof that the Dedekind zeta function of a number field L has infinitely many nontrivial zeros of multiplicity at least 2 if L has a subfield K that is a nonabelian Galois extension. Moreover, we extend this result to zeros of order 3 when the Galois group Gal(L/K) has an irreducible representation of degree at least 3, as predicted by the Artin holomorphy conjecture.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics
Oanh Nguyen, Van Vu
Summary: We investigate the local distribution of roots of random functions involving independent random variables and arbitrary analytic functions. We develop a robust framework that reduces the calculation of root distribution and interaction to cases where the random variables follow a Gaussian distribution. Our framework has applications in various models of random functions and provides a unified treatment for all of them. We derive important results, such as local universality for random trigonometric polynomials and optimal error estimates for random algebraic polynomials.
AMERICAN JOURNAL OF MATHEMATICS
(2022)
Article
Mathematics
Zoltan Buczolich, Gunther Leobacher, Alexander Steinicke
Summary: We construct a Holder continuous function on the unit interval that coincides with every function of total variation smaller than 1 passing through the origin at uncountably many points. We introduce the concept of impermeable graph and provide examples of functions with both permeable and impermeable graphs. We also demonstrate that typical continuous functions have permeable graphs. Another major result is the construction of a continuous function on the unit interval that coincides with every function of total variation smaller than 1 passing through the origin in a set of Hausdorff dimension 1.
MATHEMATISCHE NACHRICHTEN
(2023)
Article
Mathematics, Applied
Daniel Hu, Ikuya Kaneko, Spencer Martin, Carl Schildkraut
Summary: In this article, a new unconditional proof is provided for the existence of infinitely many nontrivial zeros of multiplicity at least 2 in the Dedekind zeta function of a number field, when the number field has a subfield that is a nonabelian Galois extension. Furthermore, it is extended to zeros of order 3 when the Galois group has an irreducible representation of degree at least 3, as conjectured by the Artin holomorphy conjecture.
PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
(2022)
Article
Mathematics, Applied
Elchin Hasanalizade, Quanli Shen, Peng-Jie Wong
Summary: This paper provides an explicit bound for the number of zeros of the Dedekind zeta function of a number field K and improves previous results. The improvement is based on recent work on counting zeros of Dirichlet L-functions.
MATHEMATICS OF COMPUTATION
(2022)
Article
Mathematics
Reem Alzahrani, Saiful R. Mondal
Summary: This paper aims to construct inequalities of the Redheffer type for functions defined by infinite product involving zeroes. The proofs rely on classical results regarding the monotonicity of the ratio of differentiable functions. Special cases lead to examples involving special functions such as Bessel, Struve, and Hurwitz functions, as well as other trigonometric functions.
Article
Mathematics, Applied
Ilija Tanackov, Zeljko Stevic
Summary: Newton's identities reveal the properties of multiple zeta functions of infinite polynomials with complex-conjugate roots. In a special case, the multiple zeta function is zero, including all non-trivial zeros. The general value of the infinite multiple zeta function is calculated based on Vieta's rules and is related to the special case.
Article
Engineering, Environmental
D. Posa
Summary: This paper explores new classes of covariance models with novel properties, focusing on isotropic covariance functions which are important in many applied areas. The new models are more flexible than traditional ones as they can select positive or negative covariance functions based on parameter values, as well as exhibit different behaviors near the origin. The practical implications are highlighted as these simple and adaptable isotropic covariance models are useful for various case studies.
STOCHASTIC ENVIRONMENTAL RESEARCH AND RISK ASSESSMENT
(2023)
Article
Physics, Mathematical
Federico Camia, Jianping Jiang, Charles M. M. Newman
Summary: In this paper, Ising models with ferromagnetic pair interactions are considered. The authors prove that the Ursell functions u(2k) are increasing in each interaction. As an application, a conjecture made by Nishimori and Griffiths in 1983 about the partition function of the Ising model with a complex external field is proven: the nearest zero to the origin (in the variable h) moves towards the origin as any interaction increases.
COMMUNICATIONS IN MATHEMATICAL PHYSICS
(2023)
Article
Mathematics, Applied
Di Liu, Alexandru Zaharescu
Summary: In this paper, a new type of race is introduced where two Dirichlet L-functions and test functions are analyzed and compared for large values of T.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2021)
